Kanonische Quantisierung einer regularisierten Wellengleichung bei neuer Antiteilchenvorstellung

Abstract

According to the new concept of antiparticles, the energy is equal to the modulus of the frequency, and the eigenvalues of the Schrödinger operators are primarily frequencies. Orthodox canonical quantization of appropriately chosen classical Lagrangian densities, i.e. in a Hubert space of positive definite metric, is then found to yield finite results without the use of an artifice. In the present investigation we apply the new concepts to a “realistic regularized” spinor field equation derived from a Schrödinger operator formally related to that of Heisenberg's field theory. In this case, canonical quantization leads to wave equations in the configuration space. The socalled mathematical vacuum is the same as the physical vacuum, and the bare particles are real ones. The two-particle-problem can be integrated in full rigour. In particular the binding and scattering of two particles is investigated. The latter includes a process comparable with the muon decayμ ⁻→e ⁻ v¯v, as is to be expected on the grounds of crossing symmetry. It remains to be seen how far the wave equation, introduced in an ad hoc manner, represents more than just a model. — Undoubtedly it describes a possible relativistic world, in fact a world for which we can rigorously derive nontrivial results. For some special processes binding energies, cross-sections and decay-constants are calculated. On the basis of the new concepts, the vacuum in relativistic quantum theory is independent of the interaction, just as in the nonrelativistic theory. This ought to simplify the inclusion of new symmetry properties. — Although the large number of Lagrangian densities is considerably reduced by the requirement of regularization, in a similar manner as by imposing symmetry restrictions, the principles of quantum-mechanics together with this new concept of antiparticles, just as in connection with the old concept, yield not only the particular theory developed in this context, but a general framework which we may expect to cover any quantumtheoretical relativistic problem.